What is PROPERTIES AND SKETCHING OF ROOT LOCUS

Root-Locus and Its Properties

An unstablesystemcannot perform the required control task. Now if we consider problems which are in the design of thesystemof the problem so it is corrected by the adjustment done in its closed looppoles. Although there is a technique Routh and Hurwitz criteria but when the degree of polynomial is increased so to handle it is tough there is a technique is designed which is theRoot locustechnique. This is a graph based technique which plots the locus ofpolesof thecontrol systemwith the s- parameters varied over the complete range of values.As there is a change in thegain, thesystempolesand thesystemzerosactually move around in the s-plane. These become difficult when we need to solve the higher-order equations every time for the different newgainvalue. This is the main reason to use the techniqueRoot locus. TheRoot locushelp us to graph the locations of thesystem polesandsystem zerosfor the each and every new value of thegain, it is done by following the several simple rules.

There is a closed-loopcontrol systemand thetransfer functionfor that particularsystemis defined by:

Here N is the numerator polynomial of thetransfer functionand D is the denominator polynomial of thetransfer functions. For thepolesof the equation the denominator is set to 0, and solves the characteristic equation. The locations of thepolesof a specifictransfer functionequation must satisfy the following relationship:

D(s) = 1 +KG(s)H(s) = 0

1 +KG(s)H(s) = 0

KG(s)H(s) = − 1

And finally now converting to the polar coordinates:

Now there are two equations for the locations of thepolesof asystemfor all and differentgainvalues:

The Root-Locus Procedure

There are 11 rules for a Root locus design

1. There is one branch for every root of denominator.

2. These are poles for open looptransfer function.

3. Roots of numerator are zeros and they should be less than poles.

4. A point on the real axis of the plot is part ofroot locusif it is to the left of an odd number of poles and zeros.

5. Thegainis given by

6. The design is symmetric about the real axis and complex conjugate.

7. The roots meet at real axis and break away from the axis at break away point.

If s→

Differentiated to find the maximum

8. Break away point is separated by the angles.

9. Rootloci follow the asymptotes that intersect the real axis. Asymptotes starts from

10.TheRoot locusbranch cuts the imaginary angle at 180 degree.

11. The angles of the branches makes by the complex conjugatepolesandzerosare determined by the angle equation.

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